One Variable Statistics

In grades 6 to 8, students were introduced to data sets and different ways to represent data (histograms, dot plots, box plots). Statistics is introduced as a tool to answer questions about a population that have variability in the answer. Students learn about measures of center (median, mean) and measures of variability (interquartile range, mean absolute deviation), using them to draw informal comparative inferences about two populations. Students build on and expand their understandings of statistics in this unit. The key characteristics (measures of shape, center, and spread) are again seen and in addition, students may further describe the shape of a data distribution (symmetric, skewed, flat or bell shaped) and summarize by a statistic measuring center and a statistic measuring spread. Instead of creating representations of data, the emphasis in high school is on interpreting representations and judiciously interpreting measures of center and spread. Students develop more precise understanding of measures of center. They learn that mean and median are equal for symmetrical distributions, explain why mean and median are not equal in examples of skewed distributions, select median as the better measure of center for skewed distributions, and make generalizations about what kinds of distributions have means larger than medians and which have medians larger than means. Students learn that standard deviation is a measure of spread, that a larger standard deviation means the data are more variable or spread out, and the meaning of standard deviation as “typical distance from the mean” for a symmetrical distribution. To aid in developing their understanding, students will calculate a standard deviation by hand for a small data set at least once. Given different visual representations of data (box plots, histograms, dot plots) students draw and justify significant and meaningful conclusions about the given situation. (All of these representations are frequency graphs, however, the Standards do not require students to know or use the term “frequency graph,” although they use the term “frequency tables.”) Students are introduced to two-way frequency tables and understand how to interpret relative frequencies in the context of the data represented in the tables.

In unit M1.3 on bivariate statistics (which could take place either before or after this unit), students also build their statistics foundation by learning ways to determine whether two sets of data are correlated, and how strongly. Students identify linear association and interpret slope and intercept in the context of the data. Given different visual representations of data (linear models) students draw and justify significant and meaningful conclusions about the given situation. Students begin to use technology as a means to plot data and generate correlation coefficients. In unit M2.1 on probability, students revisit two-way frequency tables from a probability standpoint, and use them as a tool for conceptualizing and finding conditional probabilities. In unit M3.6 on inferences, students combine the ideas of distributions and probability. They learn about normal distributions and use them to solve problems, and use the distributions of probability models to find the likelihood of a particular outcome. In doing so, students build on their experience with standard deviations from S1, calculating standard deviations using technology, and interpreting the results. Every high school statistics and probability standard is a modeling standard, hence modeling pervades the four units.